Types of the golden ratio. What is the golden ratio (proportion)

The golden ratio is a universal manifestation of structural harmony. It is found in nature, science, art - in everything that a person can come into contact with. Once having become acquainted with the golden rule, humanity no longer betrayed it.

DEFINITION

The most comprehensive definition of the golden ratio states that the smaller part relates to the larger, as the larger part relates to the whole. Its approximate value is 1.6180339887. In a rounded percentage value, the proportions of the parts of the whole will correspond as 62% to 38%. This relationship operates in the forms of space and time.

The ancients saw the golden ratio as a reflection of cosmic order, and Johannes Kepler called it one of the treasures of geometry. Modern science is considering golden ratio as “asymmetrical symmetry,” calling it in a broad sense a universal rule that reflects the structure and order of our world order.

STORY

The ancient Egyptians had an idea about the golden proportions, they knew about them in Rus', but for the first time the golden ratio was scientifically explained by the monk Luca Pacioli in the book “Divine Proportion” (1509), illustrations for which were supposedly made by Leonardo da Vinci. Pacioli saw in the golden section the divine trinity: the small segment personified the Son, the large segment the Father, and the whole the Holy Spirit.

The name of the Italian mathematician Leonardo Fibonacci is directly associated with the golden ratio rule. As a result of solving one of the problems, the scientist came up with a sequence of numbers now known as the Fibonacci series: 1, 2, 3, 5, 8, 13, 21, 34, 55, etc. Kepler drew attention to the relationship of this sequence to the golden proportion: “It is arranged in such a way that the two lower terms of this never-ending proportion add up to the third term, and any two last terms, if added, give the next term, and the same proportion is maintained ad infinitum " Now the Fibonacci series is the arithmetic basis for calculating the proportions of the golden section in all its manifestations.

Leonardo da Vinci also devoted a lot of time to studying the features of the golden ratio; most likely, the term itself belongs to him. His drawings of a stereometric body formed by regular pentagons prove that each of the rectangles obtained by section gives the aspect ratio in the golden division.

Over time, the golden ratio rule became an academic routine, and only the philosopher Adolf Zeising gave it a second life in 1855. He brought the proportions of the golden section to the absolute, making them universal for all phenomena of the surrounding world. However, his “mathematical aesthetics” caused a lot of criticism.

NATURE

Even without going into calculations, the golden ratio can be easily found in nature. So, the ratio of the tail and body of a lizard, the distances between the leaves on a branch fall under it, there is a golden ratio in the shape of an egg, if a conditional line is drawn through its widest part.

The Belarusian scientist Eduard Soroko, who studied the forms of golden divisions in nature, noted that everything growing and striving to take its place in space is endowed with the proportions of the golden section. In his opinion, one of the most interesting shapes This is a spiral twist.

Archimedes, paying attention to the spiral, derived an equation based on its shape, which is still used in technology. Goethe later noted nature’s attraction to spiral forms, calling the spiral the “curve of life.” Modern scientists have found that such manifestations of spiral forms in nature as a snail shell, the arrangement of sunflower seeds, spider web patterns, the movement of a hurricane, the structure of DNA and even the structure of galaxies contain the Fibonacci series.

HUMAN

Fashion designers and clothing designers make all calculations based on the proportions of the golden ratio. Man is a universal form for testing the laws of the golden ratio. Of course, by nature, not all people have ideal proportions, which creates certain difficulties with the selection of clothes.

In Leonardo da Vinci's diary there is a drawing of a naked man inscribed in a circle, in two superimposed positions. Based on the research of the Roman architect Vitruvius, Leonardo similarly tried to establish the proportions of the human body. Later, the French architect Le Corbusier, using Leonardo’s “Vitruvian Man,” created his own scale of “harmonic proportions,” which influenced the aesthetics of 20th-century architecture.

Adolf Zeising, studying the proportionality of a person, did a colossal job. He measured about two thousand human bodies, as well as many ancient statues, and concluded that the golden ratio expresses the average statistical law. In a person, almost all parts of the body are subordinate to it, but the main indicator of the golden ratio is the division of the body by the navel point.
As a result of measurements, the researcher found that the proportions of the male body 13:8 are closer to the golden ratio than the proportions female body – 8:5.

ART OF SPATIAL FORMS

The artist Vasily Surikov said “that in composition there is an immutable law, when in a picture you cannot remove or add anything, you cannot even add an extra point, this is real mathematics.” For a long time artists followed this law intuitively, but after Leonardo da Vinci, the process of creating a painting can no longer be accomplished without solving geometric problems. For example, Albrecht Durer used the proportional compass he invented to determine the points of the golden section.

Art critic F.V. Kovalev, having examined in detail the painting by Nikolai Ge “Alexander Sergeevich Pushkin in the village of Mikhailovskoye,” notes that every detail of the canvas, be it a fireplace, a bookcase, an armchair or the poet himself, is strictly inscribed in golden proportions.

Researchers of the golden ratio tirelessly study and measure architectural masterpieces, claiming that they became such because they were created according to the golden canons: their list includes the Great Pyramids of Giza, Notre Dame Cathedral, St. Basil's Cathedral, and the Parthenon.

And today, in any art of spatial forms, they try to follow the proportions of the golden section, since, according to art critics, they facilitate the perception of the work and form an aesthetic feeling in the viewer.

WORD, SOUND AND FILM

The forms of temporary art in their own way demonstrate to us the principle of the golden division. Literary scholars, for example, have noticed that the most popular number of lines in poems of the late period of Pushkin’s work corresponds to the Fibonacci series - 5, 8, 13, 21, 34.

The rule of the golden section also applies in individual works of the Russian classic. Thus, the climax of “The Queen of Spades” is the dramatic scene of Herman and the Countess, ending with the death of the latter. The story has 853 lines, and the climax occurs on line 535 (853:535 = 1.6) - this is the point of the golden ratio.

Soviet musicologist E.K. Rosenov notes the amazing accuracy of the golden ratio ratios in the strict and free forms of the works of Johann Sebastian Bach, which corresponds to the thoughtful, concentrated, technically verified style of the master. This is also true of the outstanding works of other composers, where the most striking or unexpected musical solution usually occurs at the golden ratio point.

Film director Sergei Eisenstein deliberately coordinated the script of his film “Battleship Potemkin” with the rule of the golden ratio, dividing the film into five parts. In the first three sections the action takes place on the ship, and in the last two - in Odessa. The transition to scenes in the city is the golden middle of the film.

Bibliographic description: Maksimenko O. V., Pastor V. S., Vorfolomeeva P. V., Mozikova K. A., Nikolaeva M. E., Shmeleva O. V. To the concept of the Golden Section // Young scientist. 2016. No. 6.1. P. 35-39..03.2019).



“Geometry has two treasures:

one of them - Pythagorean theorem,

another is the division of a segment in the average and extreme ratio"

Johannes Kepler

Keywords: golden ratio, golden proportions, scientific phenomenon.

The purpose of our work is to study sources of information relating to the “Golden Ratio” in various areas knowledge, identifying patterns and finding connections between sciences, identifying the practical meaning of the Golden Ratio.

The relevance of this study is determined by the centuries-old history of the use of the golden ratio in mathematics and art. What the ancients puzzled over remains relevant and arouses the interest of contemporaries.

At all times, people have tried to find patterns in the world around them. They surrounded themselves with objects of the “correct” form from their point of view. Only with the development of mathematics did people manage to measure the “golden ratio”, which later became known as the “Golden Section”.

Golden ratio- harmonic proportion

The golden ratio is such a proportional division of a segment into unequal parts, in which the entire segment is related to the larger part as the larger part itself is related to the smaller one; or, in other words, the smaller segment is related to the larger one as the larger one is to the whole (Fig. 1).

a: b = b: c

Rice. 1. Division of a segment according to golden proportions

Let us remind you what the golden ratio is. The most comprehensive definition of the golden ratio states that the smaller part is related to the larger one, as the larger part is to the whole. Its approximate value is 1.6180339887. In a rounded percentage value, the proportions of the parts of the whole will correspond as 62% to 38%. This relationship operates in the forms of space and time.

Golden Triangle andrectangle

In addition to dividing a segment into unequal parts (golden ratio), golden triangle and golden rectangle are considered.

A golden rectangle is a rectangle whose side lengths are in the golden proportion (Fig. 2).

Each end of the pentagonal star represents a golden triangle. Its sides form an angle of 36° at the apex, and the base, laid on the side, divides it in the proportion of the golden ratio (Fig. 3).

Fig.2. golden rectangle

Fig.3 Golden triangle

Pentacle

In a regular five-pointed star, each segment is divided by a segment intersecting it in the golden ratio, i.e. the ratio of the blue segment to the green, red to blue, green to violet is 1.618 (Fig. 4).

Fig.4. Pentagram-hygiea

Pythagoras argued that the pentagram, or, as he called it, hygieia, represents mathematical perfection, since it hides the golden ratio. The ratio of the blue segment to green, red to blue, green to violet is the golden ratio.

Fibonacci series

The series of numbers 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, etc. is known as the Fibonacci series. The peculiarity of the sequence of numbers is that each of its members, starting from the third, equal to the sum two previous, and the ratio of adjacent numbers in the series approaches the ratio of the golden division.

So, 21:34 = 0.617

34: 55 = 0,618.

History of the golden ratio

It is generally accepted that the concept of the golden division was introduced into scientific use by Pythagoras, ancient Greek philosopher and mathematician (VI century BC). There is an assumption that Pythagoras borrowed his knowledge of the golden division from the Egyptians and Babylonians. Indeed, the proportions of the Cheops pyramid, temples, bas-reliefs, household items and jewelry from the tomb of Tutankhamun indicate that Egyptian craftsmen used the ratios of the golden division when creating them.

Golden proportions inparts of the human body

In 1855, the German researcher of the golden ratio, Professor Zeising, published his work “Aesthetic Studies”.

Zeising measured about two thousand human bodies and came to the conclusion that the golden ratio expresses the average statistical law (Fig. 5).

Fig.5 Golden proportions in parts of the human body

Golden ratio inwildlife

It's amazing how just one mathematical concept appears in many sections human knowledge. It seems to permeate everything in the world, connecting harmony and chaos, mathematics and art.

IN biological research it was shown that, starting with viruses and plants and ending with the human body, the golden proportion is revealed everywhere, characterizing the proportionality and harmony of their structure. The golden ratio is recognized as a universal law of living systems.

At first glance, the lizard has proportions that are pleasing to our eyes - the length of its tail is related to the length of the rest of the body as 62 to 38 (Fig. 6).

Fig. 6 Golden proportions in parts of the lizard’s body

Golden ratio inarchitecture

In books about the “golden ratio” you can find the remark that in architecture, as in painting, everything depends on the position of the observer, and if some proportions in a building from one side seem to form the “golden ratio”, then from other points of view they will look different. The “Golden Ratio” gives the most relaxed ratio of the sizes of certain lengths.

One of the most beautiful works of ancient Greek architecture is the Parthenon (Fig. 7). The ratio of the building's height to its length is 0.618. If we divide the Parthenon according to the “golden section”, we will get certain protrusions of the facade.

Another example from ancient architecture is the Cheops pyramid (Fig. 8).

Proportions Great Pyramid aged in the "Golden Ratio"

The ancient builders managed to erect this majestic monument with almost perfect engineering precision and symmetry.

Fig.7. Parthenon

Fig.8. The Pyramid of Cheops

Golden ratio insculpture

The proportions of the “golden section” create the impression of harmony of beauty, which is why sculptors used them in their works. For example, the famous statue of Apollo Belvedere consists of parts divided according to golden ratios (Fig. 9).

Fig.9 Statue of Apollo Belvedere

Golden ratio inpainting

Moving on to examples of the “golden ratio” in painting, one cannot help but focus on the work of Leonardo da Vinci. Let's look carefully at the painting "La Gioconda". The composition of the portrait is built on golden triangles (Fig. 10).

Fig. 10 Leonardo da Vinci “La Gioconda”

Another example of the golden ratio in painting is Raphael’s painting “Massacre of the Innocents” (Fig. 11). In Raphael's preparatory sketch, red lines are drawn running from the semantic center of the composition. If you naturally connect these pieces with a curved dotted line, then with very great accuracy you get... a golden spiral!

Fig. 11. Raphael "Massacre of the Innocents"

Golden ratio inliterary works

The forms of temporary art in their own way demonstrate to us the principle of the golden division. The rule of the golden section also applies in individual works of the Russian classic. So, in the story “The Queen of Spades” there are 853 lines, and the climax occurs on line 535 (853:535 = 1.6) - this is the point of the golden ratio.

Golden ratio infilms

Film director Sergei Eisenstein deliberately coordinated the script of his film “Battleship Potemkin” with the rule of the golden ratio, dividing the film into five parts.

Conclusion

The golden ratio was known back in ancient Egypt and Babylon, in India and China. The great Pythagoras created a secret school where the mystical essence of the “golden ratio” was studied. Euclid used it when creating his geometry, and Phidias - his immortal sculptures. Plato said that the Universe is arranged according to the “golden ratio”. And Aristotle found a correspondence between the “golden ratio” and the ethical law. The highest harmony of the “golden ratio” will be preached by Leonardo da Vinci and Michelangelo, because beauty and the “golden ratio” are one and the same thing. And Christian mystics will draw pentagrams of the “golden ratio” on the walls of their monasteries, fleeing from the Devil. At the same time, scientists - from Pacioli to Einstein - will search, but will never find its exact meaning. An endless series after the decimal point - 1.6180339887... A strange, mysterious, inexplicable thing: this divine proportion mystically accompanies all living things. Inanimate nature doesn’t know what the “golden ratio” is. But you will certainly see this proportion in the curves of sea shells, and in the shape of flowers, and in the appearance of beetles, and in the beautiful human body. Everything living and everything beautiful - everything obeys the divine law, whose name is the “golden ratio”. So what is the “golden ratio”? What is this perfect, divine combination? Maybe this is the law of beauty? Or is he still a mystical secret? Scientific phenomenon or ethical principle? The answer is still unknown. More precisely - no, it is known. The “golden ratio” is both, and the third. Only not separately, but simultaneously... And this is his true mystery, his great secret.

Literature:

1. Vilenkin N. Ya., Zhokhov V. I. and others. Mathematics - 6. - M.: Mnemosyne, 2015
2. Korbalan F. The Golden Ratio. The mathematical language of beauty. (World of Mathematics Vol.1). - M.: DeAgostini, 2014
3. Timerding G. E. Golden ratio. - M.: Librocom, 2009

Keywords: golden ratio, golden proportions, scientific phenomenon.

Annotation: The golden ratio is a universal manifestation of structural harmony. It is found in nature, science, art - in everything that a person can come into contact with. The authors of the article examine the literature, find connections between sciences related to the Golden Ratio, and identify the practical meaning of the golden proportions.

Golden ratio- this is such a proportional division of a segment into unequal parts, in which the smaller segment is related to the larger one, as the larger one is to the whole.

a: b = b: c or c: b = b: a.

This proportion is:

For example, in a regular five-pointed star, each segment is divided by a segment intersecting it in the golden ratio (i.e., the ratio of the blue segment to the green, red to blue, green to violet is equal 1.618

It is generally accepted that the concept of the golden ratio was introduced into scientific use by Pythagoras. There is an assumption that Pythagoras borrowed his knowledge from the Egyptians and Babylonians. Indeed, the proportions of the Cheops pyramid, temples, bas-reliefs, household items and jewelry from the tomb of Tutankhamun indicate that Egyptian craftsmen used the ratios of the golden division when creating them.

In 1855, the German researcher of the golden ratio, Professor Zeising, published his work "Aesthetic Research".
Zeising measured about two thousand human bodies and came to the conclusion that the golden ratio expresses the average statistical law.

Golden proportions in parts of the human body

The division of the body by the navel point is the most important indicator of the golden ratio. The proportions of the male body fluctuate within the average ratio of 13: 8 = 1.625 and are somewhat closer to the golden ratio than the proportions of the female body, in relation to which the average value of the proportion is expressed in the ratio 8: 5 = 1.6.

In a newborn the proportion is 1:1, by the age of 13 it is 1.6, and by the age of 21 it is equal to that of a man.
The proportions of the golden ratio also appear in relation to other parts of the body - the length of the shoulder, forearm and hand, hand and fingers, etc.
Zeising tested the validity of his theory on Greek statues. He developed the proportions of Apollo Belvedere in the most detail. Greek vases and architectural structures were examined different eras, plants, animals, bird eggs, musical tones, poetic meters.

Zeising gave a definition to the golden ratio and showed how it is expressed in straight line segments and in numbers. When the figures expressing the lengths of the segments were obtained, Zeising saw that they amounted to Fibonacci series.

A series of numbers 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, etc. known as the Fibonacci series. The peculiarity of the sequence of numbers is that each of its members, starting from the third, equal to the sum of the previous two 2 + 3 = 5; 3 + 5 = 8; 5 + 8 = 13, 8 + 13 = 21; 13 + 21 = 34, etc., and the ratio of adjacent numbers in the series approaches the ratio of the golden division.

So, 21: 34 = 0.617, and 34: 55 = 0,618. (or 1.618 , if divided larger number to less).

Fibonacci series could have remained only a mathematical incident, if not for the fact that all researchers of the golden division in the plant and animal world, not to mention art, invariably came to this series as an arithmetic expression of the law of the golden section.

Golden ratio in art

Back in 1925, art critic L.L. Sabaneev, having analyzed 1,770 musical works by 42 authors, showed that the vast majority of outstanding works can be easily divided into parts either by theme, or by intonation structure, or by modal structure, which are in relation to each other golden ratio.

Moreover, the more talented the composer, the more golden sections are found in his works. In Arensky, Beethoven, Borodin, Haydn, Mozart, Scriabin, Chopin and Schubert, golden sections were found in 90% of all works. According to Sabaneev, the golden ratio leads to the impression of a special harmony of a musical composition.

In cinema, S. Eisenstein artificially constructed the film Battleship Potemkin according to the rules of the “golden ratio”. He broke the tape into five parts. In the first three, the action takes place on a ship. In the last two - in Odessa, where the uprising is unfolding. This transition to the city occurs exactly at the golden ratio point. And each part has its own fracture, which occurs according to the law of the golden ratio.

One of the most beautiful works of ancient Greek architecture is the Parthenon (5th century BC).

The figures show a number of patterns associated with the golden ratio. The proportions of the building can be expressed through various powers of the number Ф=0.618...

On the floor plan of the Parthenon you can also see the “golden rectangles”:

We can see the golden ratio in the building of Notre Dame Cathedral (Notre Dame de Paris) and in the Pyramid of Cheops:

Not only the Egyptian pyramids were built in accordance with the perfect proportions of the golden ratio; the same phenomenon was found in the Mexican pyramids.

The golden proportion was used by many ancient sculptors. The golden proportion of the statue of Apollo Belvedere is known: the height of the depicted person is divided by the umbilical line in the golden section.

Moving on to examples of the “golden ratio” in painting, one cannot help but focus on the work of Leonardo da Vinci. Let's look closely at the painting "La Gioconda". The composition of the portrait is based on “golden triangles”.

Golden ratio in fonts and household items

Golden ratio in nature

Biological studies have shown that, starting with viruses and plants and ending with the human body, the golden proportion is revealed everywhere, characterizing the proportionality and harmony of their structure. The golden ratio is recognized as a universal law of living systems.

It was found that number series Fibonacci numbers characterize structural organization many living systems. For example, the helical leaf arrangement on a branch forms a fraction (number of revolutions on the stem/number of leaves in a cycle, eg 2/5; 3/8; 5/13), corresponding to the Fibonacci series.

The “golden” proportion of five-petal flowers of apple, pear and many other plants is well known. Carriers genetic code- DNA and RNA molecules - have a double helix structure; its dimensions almost completely correspond to the numbers of the Fibonacci series.

Goethe emphasized nature's tendency toward spirality.

The spider weaves its web in a spiral pattern. A hurricane is spinning like a spiral. Frightened herd reindeer spirals away.

Goethe called the spiral the “curve of life.” The spiral was seen in the arrangement of sunflower seeds, pine cones, pineapples, cacti, etc.

Flowers and seeds of sunflowers, chamomiles, scales in pineapple fruits, conifer cones are “packed” in logarithmic (“golden”) spirals, curling towards each other, and the numbers of “right” and “left” spirals are always related to each other, like neighboring numbers Fibonacci.

Consider a chicory shoot. A shoot has formed from the main stem. The first leaf was located right there. The shoot makes a strong ejection into space, stops, releases a leaf, but this time it is shorter than the first one, again makes an ejection into space, but with less force, releases a leaf of an even smaller size and is ejected again.

If the first emission is taken as 100 units, then the second is equal to 62 units, the third – 38, the fourth – 24, etc. The length of the petals is also subject to the golden proportion. In growing and conquering space, the plant maintained certain proportions. The impulses of its growth gradually decreased in proportion to the golden ratio.

In many butterflies, the ratio of the sizes of the thoracic and abdominal parts of the body corresponds to the golden ratio. Folding my wings moth forms a regular equilateral triangle. But if you spread your wings, you will see the same principle of dividing the body into 2,3,5,8. The dragonfly is also created according to the laws of the golden proportion: the ratio of the lengths of the tail and body is equal to the ratio of the total length to the length of the tail.

In a lizard, the length of its tail is related to the length of the rest of the body as 62 to 38. You can notice the golden proportions if you look closely at a bird's egg.

Airbrushing is based on the same "pillars" as other forms of art.

Our entire world can be described by numbers. Many numbers play such a significant role in this description that they have their own names: Pi, exponent (e), etc. Among these “nominal” numbers there is something quite remarkable. Mathematicians, artists, architects in different times they called it “golden number”, “divine number”, “divine section”. The term “golden ratio” was coined by Claudius Ptolemy, and it became popular thanks to Leonardo Da Vinci, who used it extensively in his works. People of art have noticed that the proportions of forms that are especially pleasing to the eye for perception are based on the “golden ratio”.

So what is this number? The golden ratio is the number Phi (Phi) equal to 1.61803. The number is named after the great ancient Greek sculptor Phidias, who used it in his sculptures. How to clearly demonstrate the principle of the “golden ratio”? Let's give a simple example. If you build a rectangle, one side of which is 1.618 times longer than the other, then the resulting aspect ratio is the “golden ratio”. The most common "golden rectangles" in modern world- these are credit cards. The human body is considered beautiful, and its proportions are considered ideal, if the ratio between the smaller and larger parts of the body is equal to the ratio between the larger part and the whole, that is, equal to the number Phi.

***
The most famous mathematical work of ancient science is Euclid's Elements. It was from the “Principles” that the geometric problem “on the division of a segment in extreme and mean ratio” came to us. Which is the “Golden Ratio” itself.
The essence of the task is this:
Let us divide the segment AB by point C in such a ratio that the larger part of the segment CB is related to the smaller part of the segment AC as the segment AB is to its larger part CB, i.e.

Let us denote proportion (1.1) by x. Then, taking into account that AB = AC + CB, proportion (1.1) can be written in the following form:

This gives us the following algebraic equation for calculating the required proportion x:

X* = x + 1. (1.2)
x* - squared

From the “physical meaning” of proportion (1.1) it follows that the desired solution to equation (1.2) must be a positive number, from which it follows that the solution to the problem of dividing a segment in extreme and mean ratio is the positive root of equation (1.2), which we denote by , that is

The approximate value of the golden ratio is:
= 1,61803 39887 49894 84820 45868 34365 63811 77203…

GOLDEN GEOMETRIC FIGURES

Based on the above proportions in geometry, the following concepts of gold are defined: geometric shapes:
- golden rectangle (in which the ratio of the larger side to the smaller side is equal to the golden ratio);
- gold right triangle;
- golden ellipse;
- golden isosceles triangle.

A right triangle with sides 3:4:5 is called "perfect", "sacred" or "Egyptian".
Creators Egyptian pyramids chose the golden right triangle as the “main geometric idea” for the Cheops pyramid, and the “sacred” triangle for the Khafre pyramid.

Pentagon (“pentagonon” - Greek), regular pentagon. If we draw all the diagonals in the pentagon, the result is a pentagonal star called a pentagram (“pentagrammon” - Greek: “pente” - five and “grammon” - line) or pentacle.

The pentagram, called folk beliefs"witch's foot", played a large role in all magical sciences and was considered as a means of protection against evil spirits.
Every eight years the planet Venus describes an absolutely correct pentacle in big circle celestial sphere.
The Pentagon building, the US military department, is shaped like a Pentagon.

The Pentagon and Pentacle include a number of remarkable figures that have been widely used in works of art. The so-called law of the golden cup, which was used by ancient sculptors and goldsmiths, is widely known in ancient art. The shaded portion of the pentagon gives a schematic representation of the golden cup.

Once upon a time in the Soviet Union there was a State Quality Mark, in which the motifs of the golden cup are clearly visible.

In living nature, forms based on pentagonal symmetry are widespread - sea ​​stars, sea ​​urchins, flowers..

HARMONY OF THE GOLDEN RATIO
(short review art history)

The great works of Greek sculptors: Phidias, Polyctetus, Myron, Praxiteles have long been rightfully considered the standard of beauty of the human body, an example of a harmonious physique. In their creations, Greek masters used the principle of the golden proportion. One of the highest achievements of classical Greek art is the statue of Doryphoros, sculpted by Polyctetus in the 5th century BC. e. This statue is considered the best example for analyzing the proportions of the ideal human body, established by ancient Greek sculptors, and is directly related to the Golden Ratio. M=0.618…
Venus de Milo, goddess Aphrodite statue and standard female beauty, is one of the best monuments of Greek sculptural art.

Leonardo Da Vinci used the Golden Ratio proportions in many of his most famous works, most notably The Last Supper and the famous La Gioconda.
Researchers of the painting “La Gioconda” discovered that the compositional structure of the painting is based on two golden triangles, their bases facing each other. Harmonic analysis of the picture shows that the pupil of the left eye, through which the vertical axis of the canvas passes, is located at the intersection of two bisectors of the upper golden triangle, which, on the one hand, bisect the angles at the base of the golden triangle, and on the other hand, at the points of intersection with the hips of the golden triangle triangles divide them in proportion to the Golden Ratio. Thus, Leonardo Da Vinci used in his painting not only the principle of symmetry, but also the Golden Ratio.

The painting “The Holy Family” by Michelangelo is recognized as one of the masterpieces of Western European art of the Renaissance. Harmonic analysis showed that the composition of the painting is based on a pentacle.

The proportions of the statue of David (by Michelangelo) are based on the Golden Ratio.

A striking example Baroque architecture, the Smolny Cathedral in St. Petersburg, makes an indelible impression. The Golden Ratio is also seen in its basic proportions.

In the famous painting “Ship Grove” by Ivan Shishkin, motifs of the Golden Ratio are visible. A brightly sunlit pine tree (standing in the foreground) divides the picture horizontally with the Golden Ratio. To the right of the pine tree is a hill lit by the sun. He divides the picture vertically with the Golden Ratio. To the left of the main pine tree there are many pine trees - you can continue dividing the Golden Ratio horizontally on the left side of the picture. The presence in the picture of bright verticals and horizontals, dividing it in relation to the Golden Ratio, gives it a character of balance and calm.

Construction of the UN headquarters in New York was completed in 1943. The building then attracted everyone's attention not only as a public building created using the latest architectural means, but also as the first example of the use of a continuous solar modulating screen on one of the facades. This building also displays the Golden Ratio motifs. In the composition of the building, three golden rectangles placed on top of each other clearly stand out, which are its main architectural idea.

Any piece of music has a temporal extension and is divided by certain “aesthetic milestones” into separate parts that attract attention and facilitate perception as a whole. These milestones can be the dynamic and intonation climaxes of a musical work. Separate time intervals of a musical work, connected by a “climax event,” as a rule, are in the Golden Ratio ratio. In the musical works of various composers, not just one Golden Ratio is usually stated, but a whole series of similar sections. Largest quantity works in which the Golden Ratio is present in Arensky (95%), Beethoven (97%), Haydn (97%), Mozart (91%), Scriabin (90%), Chopin (92%), Schubert (91%) .

If music is the harmonic ordering of sounds, then poetry is the harmonic ordering of speech. A clear rhythm, a natural alternation of stressed and unstressed syllables, an ordered meter of poems, and their emotional richness make poetry sister musical works. The golden ratio in poetry primarily manifests itself as the presence certain point poem (climax, semantic turning point, main idea work) in a line falling on the point of division of the total number of lines of the poem in the golden ratio. So, if a poem contains 100 lines, then the first point of the Golden Ratio falls on the 62nd line (62%), the second on the 38th (38%), etc. The works of Alexander Sergeevich Pushkin, including “Eugene Onegin" - the finest correspondence to the golden proportion! Works by Shota Rustaveli and M.Yu. Lermontov are also built according to the principle of the Golden Section.

One of modern species art - cinema, which incorporates dramaturgy of action, painting, music. It is right to look for manifestations of the Golden Ratio in outstanding works of cinema. The first to do this was the creator of the world cinema masterpiece “Battleship Potemkin,” film director Sergei Eisenstein. In constructing this picture, he managed to embody the basic principle of harmony - the Golden Ratio. As Eisenstein himself notes, the red flag on the mast of the mutinous battleship (the climax of the film) flies at the point of the golden ratio, counted from the end of the film.

For many millennia, the Golden Ratio has been the object of admiration and worship of outstanding scientists and thinkers: Pythagoras, Plato, Euclid, Luca Pacioli, Johannes Kepler, Pavel Florensky...
Currently, the Golden Ratio is a source of new fruitful ideas in mathematics and theoretical physics, biology and botany, economics and computer science...

The material is based on the book “The Da Vinci Code and Fibonacci Series” by A. Stakhov, A. Sluchenkova, I. Shcherbakov, published in 2007, publishing house “Peter”.

The Golden Ratio is a simple principle that can help make a design visually pleasing. In this article we will explain in detail how and why to use it.

A mathematical proportion common in nature, called the Golden Ratio, or Golden mean, is based on the Fibonacci Sequence (which you most likely heard about in school, or read about in Dan Brown's book "The Da Vinci Code"), and implies an aspect ratio of 1:1.61.

This ratio is often found in our lives (shells, pineapples, flowers, etc.) and therefore is perceived by a person as something natural and pleasing to the eye.

→ The golden ratio is the relationship between two numbers in the Fibonacci sequence
→ Plotting this sequence to scale produces the spirals that can be seen in nature.

It is believed that the Golden Ratio has been used by mankind in art and design for more than 4 thousand years, and perhaps even more, according to scientists who claim that the ancient Egyptians used this principle when building the pyramids.

Famous examples

As we have already said, the Golden Ratio can be seen throughout the history of art and architecture. Here are some examples that only confirm the validity of using this principle:

Architecture: Parthenon

In ancient Greek architecture, the Golden Ratio was used to calculate the ideal proportion between the height and width of a building, the dimensions of a portico, and even the distance between columns. Subsequently, this principle was inherited by the architecture of neoclassicism.

Art: last supper

For artists, composition is the foundation. Leonardo da Vinci, like many other artists, was guided by the principle of the Golden Ratio: in the Last Supper, for example, the figures of the disciples are located in the lower two-thirds (the larger of the two parts of the Golden Ratio), and Jesus is placed exactly in the center between two rectangles.

Web design: Twitter redesign in 2010

Twitter creative director Doug Bowman posted a screenshot on his Flickr account explaining the use of the Golden Ratio principle for the 2010 redesign. “Anyone interested in #NewTwitter proportions, know that everything was done for a reason,” he said.

Apple iCloud

The iCloud service icon is also not a random sketch. As Takamasa Matsumoto explained in his blog (original Japanese version), everything is built on the mathematics of the Golden Ratio, the anatomy of which can be seen in the picture on the right.

How to construct the Golden Ratio?

The construction is quite simple, and starts with the main square:

Draw a square. This will form the length of the “short side” of the rectangle.

Divide the square in half with a vertical line so that you get two rectangles.

In one rectangle, draw a line by joining opposite corners.

Expand this line horizontally as shown in the figure.

Create another rectangle using the horizontal line you drew in the previous steps as a guide. Ready!

"Golden" instruments

If drawing and measuring is not your thing favorite hobby, leave all the grunt work to tools that are designed specifically for this. With the help of the 4 editors below you can easily find the Golden Ratio!

The GoldenRATIO application helps you develop websites, interfaces and layouts in accordance with the Golden Ratio. It's available on the Mac App Store for $2.99, and has a built-in calculator with visual feedback, and a convenient “Favorites” function that stores settings for recurring tasks. Compatible with Adobe Photoshop.

This calculator will help you create the perfect typography for your website according to the principles of the Golden Ratio. Just enter the font size, content width in the field on the site, and click “Set my type”!

It's simple and free application for Mac and PC. Just enter a number and it will calculate the proportion for it according to the Golden Ratio rule.

A convenient program that will relieve you of the need for calculations and drawing grids. It makes finding ideal proportions easier than ever! Works with all graphic editors, including Photoshop. Despite the fact that the tool is paid - $49, it is possible to test the trial version for 30 days.